Question

Square $ABCD$ in the coordinate plane has vertices at the points $A(1, 1),$ $B(-1, 1),$ $C(-1, -1),$ and $D(1, -1).$ Consider the following four transformations:

$L,$ a rotation of $90^\circ$ counterclockwise around the origin;

$R,$ a rotation of $90^\circ$ clockwise around the origin;

$H,$ a reflection across the $x$ -axis; and

$V,$ a reflection across the $y$ -axis.

Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1, 1)$ to $(-1, -1)$ and would send the vertex $B$ at $(-1, 1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)

$2^{37}$

$3 \cdot 2^{36}$

$2^{38}$

$3 \cdot 2^{37}$

$2^{39}$

Answer: C

Concepts:transformation casework

Difficulty rating: 2000

Solution:

Label the vertices $0,1,2,3$ cyclically. Each allowed transformation has the form $j\mapsto \varepsilon j+\delta\pmod 4,$ where $\varepsilon\in\{1,-1\}$ and $\delta\in\{1,-1\}.$ These four choices give exactly the two quarter-turns and the two stated reflections.

Under composition, the parity of $\delta$ changes at every move. Thus a composition of $19$ allowed transformations again has odd $\delta,$ and so is one of the four allowed transformations. Its inverse is also allowed. Consequently every sequence of the first $19$ moves has exactly one choice for the final move, giving $4^{19}=2^{38}$ successful sequences.

Thus, the correct answer is C.

2020 AMC 12B Problem 19

Answer: C

Difficulty rating: 2000

Solution:

Problem 19 in Other Years